Description

Compute Eulerian numbers and Stirling numbers of the first and second kind, possibly vectorized for all k “at once”.

Usage

Arguments

Details

Eulerian numbers:
A(n,k) = the number of permutations of 1,2,...,n with exactly k ascents (or exactly k descents).

Stirling numbers of the first kind:
s(n,k) = (-1)^n-k times the number of permutations of 1,2,...,n with exactly k cycles.

Stirling numbers of the second kind:
S(n,k) is the number of ways of partitioning a set of n elements into k non-empty subsets.

Value

A(n,k), s(n,k) or S(n,k), respectively.

Eulerian.all(n) is the same as sapply(0:(n-1), Eulerian, n=n) (for n > 0),
Stirling1.all(n) is the same as sapply(1:n, Stirling1, n=n), and
Stirling2.all(n) is the same as sapply(1:n, Stirling2, n=n), but more efficient.

Note

For typical double precision arithmetic,
Eulerian*(n, *) overflow (to Inf) for n ≥ 172,
Stirling1*(n, *) overflow (to +/-Inf) for n ≥ 171, and
Stirling2*(n, *) overflow (to Inf) for n ≥ 220.

Author(s)

Martin Maechler ("direct": May 1992)

References

Eulerians:

NIST Digital Library of Mathematical Functions, 26.14: https://dlmf.nist.gov/26.14

Stirling numbers:

Abramowitz and Stegun 24,1,4 (p. 824-5 ; Table 24.4, p.835); Closed Form : p.824 "C."

NIST Digital Library of Mathematical Functions, 26.8: https://dlmf.nist.gov/26.8

See Also

chooseZ for the binomial coefficients.

Examples